Nearly two years have passed since I wrote my first article pitting credit card titan American Express (NYSE:AXP) against 21st-century payments processor Square (NYSE:SQ). I tagged AmEx for the win, as I felt at the time that it had the more durable and proven business model. Square was still experiencing growing pains.

Since then, much has changed for Square, if not necessarily for American Express. But has it changed enough to make it a winner in a rematch between the two stocks? Read on to find out. 

The shape of success

Square's stock has been on an absolute tear this year, making it a real standout in the finance sector. On the back of a series of very strong quarterly results, it's risen by nearly 260% since January. 

What helps greatly is that its core Reader hardware and Point of Sale software combination is becoming more popular outside its original small-business target market. And as is typical in the payment processing business, more volume means more revenue.

Square's gross payment volume for its most recently reported quarter was over $17 billion, while its adjusted revenue came in at $257 million. Those figures represented stellar year-over-year growth of 31% and 45%, respectively.

A smiling barista uses a tablet at point of sale.

Image source: Getty Images

More encouragingly, Square is expanding its business in sensible, quite clever ways. For example, in 2014 it acquired Caviar, a delivery and pickup ordering system for restaurants.

Now, that wouldn't seem to be a huge-potential business at first glance, but it helps pull more customers into the Square ecosystem. A client using Caviar for delivery and pickup would be inclined toward the Reader and Point of Sale hardware/software combination for its front end operations, and perhaps down the road consider a loan from Square Capital to expand its business.

This is working. All told, in the third quarter, the company nearly doubled its take from subscription and services-based products, the catch-all category that includes complementary offerings such as Caviar and Square Capital.

Costly divorce

Veteran AmEx continues to power along on the strength of a solid and long-established business. As ever, it's the leading "closed-loop" card payments company, meaning that it not only processes the transactions of its namesake cards, but it also issues the vast majority of them.

But the company has suffered a few body blows over the past few years, most notably its unexpected 2015 divorce from Costco Wholesale (NASDAQ:COST). For years, AmEx was not only the exclusive credit card accepted at Costco registers, but it was also the co-brander of Costco's proprietary plastic. The loss of that business, it almost goes without saying, was devastating.

The company has done a good job picking itself off the ground following that hit, and investors have been coming back to the stock. But that was a heavy blow, and the shares are still up only 16% from their level when the Costco news first broke.

American Express corporate card

Image source: Getty Images

In other words, the market still hasn't forgiven AmEx for the loss of its monster client.

That's unfair; the company has been recovering nicely. Its third-quarter results clearly showed that it's executing well on its post-Costco, two-year turnaround plan. All of its business segments grew encouragingly, and as a result revenue clocked in at $8.4 billion, 9% higher year over year -- a very good figure for a wounded giant in recovery mode. Better, net profit rose by 19% to $1.4 billion.

Those aren't the growth figures of an obsolete, lumbering company. Rather, they reveal a hungry business with something to prove.

Picking the preferred payments processor

So, in short, I like how both companies have performed lately. Square is the more compelling growth story, while AmEx is a portrait of a strong business admirably and effectively overcoming the shock of a major split-up.

Comparing the five-year forward P/E-to-growth ratio of the two reveals that Square -- which analysts collectively expect to start posting a headline net profit for fiscal 2018 -- trades at a PEG of 1.1. That beats AmEx's 1.5.

This indicates that even after its share price run-up, Square is still a bargain when matched against its older rival. 

Although some premium is justified given that AmEx is a proven business with well-seasoned management, I think Square's more attractive valuation in comparison to its growth prospects gives it an edge. And Square's starting to prove that it can compete -- its success in numerous areas of its business isn't a fluke, and it bodes very well for the future.

The combination of exciting growth story and still-attractive valuation has won me over; in contrast to my first article on this subject, I pick Square as the better stock buy for this round.

Eric Volkman has no position in any of the stocks mentioned. The Motley Fool owns shares of Square. The Motley Fool recommends American Express and Costco Wholesale. The Motley Fool has a disclosure policy.